Solved: In 45 and 46 the given figure represents the graph

In Problems 59 and 60 use a CAS to compute all derivatives and to carry out the simplifications needed to verify that the indicated function is a particular solution of the given differential equation. \(x^{3} y^{\prime \prime \prime}+2 x^{2} y^{\prime \prime}+20 x y^{\prime}-78 y=0\) ; \(y=20 \frac{\cos (5 \ln x)}{x}-3 \frac{\sin (5 \ln x)}{x}\) Text Transcription: x^3 y^prime prime prime + 2x^2 y^prime prime + 20xy^prime - 78y = 0 y=20 cos (5 ln x)/x-3 sin (5 ln x)/x

In Problems 1–8 state the order of the given ordinary differential equation. Determine whether the equation is linear or nonlinear by matching it with (6). \(\frac{d^{2} u}{d r^{2}}+\frac{d u}{d r}+u=\cos (r+u)\) Text Transcription: d^2 u/dr^2 + du/dr + u = cos (r+u)

In Problems 1–8 state the order of the given ordinary differential equation. Determine whether the equation is linear or nonlinear by matching it with (6). \(t^{5} y^{(4)}-t^{3} y^{\prime \prime}+6 y=0\) Text Transcription: t^5 y^(4) - t^3 y^prime prime + 6y = 0

In Problems 1–8 state the order of the given ordinary differential equation. Determine whether the equation is linear or nonlinear by matching it with (6). \(\frac{d^{2} y}{d x^{2}}=\sqrt{1+\left(\frac{d y}{d x}\right)^{2}}\) Text Transcription: d^2 y/dx^2 = sqrt 1+(dy/dx)^2

In Problems 1–8 state the order of the given ordinary differential equation. Determine whether the equation is linear or nonlinear by matching it with (6). \(x \frac{d^{3} y}{d x^{3}}-\left(\frac{d y}{d x}\right)^{4}+y=0\) Text Transcription: x d^3 y/dx^3 - (dy/dx)^4 + y = 0

In Problems 1–8 state the order of the given ordinary differential equation. Determine whether the equation is linear or nonlinear by matching it with (6). \((1-x) y^{\prime \prime}-4 x y^{\prime}+5 y=\cos x\) Text Transcription: (1-x) y^prime prime - 4xy^prime + 5y = cos x

In Problems 1–8 state the order of the given ordinary differential equation. Determine whether the equation is linear or nonlinear by matching it with (6). \(\frac{d^{2} R}{d t^{2}}=-\frac{k}{R^{2}}\) Text Transcription: d^2 R/dt^2 = -k/R^2

In Problems 1–8 state the order of the given ordinary differential equation. Determine whether the equation is linear or nonlinear by matching it with (6). \((\sin \theta) y^{\prime \prime \prime}-(\cos \theta) y^{\prime}=2\) Text Transcription: (sin theta) y^prime prime prime - (cos theta) y^prime = 2

In Problems 1–8 state the order of the given ordinary differential equation. Determine whether the equation is linear or nonlinear by matching it with (6). \(\ddot{x}-\left(1-\frac{\dot{x}^{2}}{3}\right) \dot{x}+x=0\) Text Transcription: ddot x - (1-dot x^2 / 3) dot x + x = 0

In Problems 9 and 10 determine whether the given first-order differential equation is linear in the indicated dependent variable by matching it with the first differential equation given in (7). \(\left(y^{2}-1\right) \ d x+x d y=0\) ; in y ; in x Text Transcription: (y^2 - 1) dx + xdy = 0

In Problems 11–14 verify that the indicated function is an explicit solution of the given differential equation. Assume an appropriate interval I of definition for each solution \(\frac{d y}{d t}+20 y=24\) ; \(y=\frac{6}{5}-\frac{6}{5} e^{-20 t}\) Text Transcription: dy/dt + 20y = 24 y = 6/5 - 6/5 e^-20t

In Problems 9 and 10 determine whether the given first-order differential equation is linear in the indicated dependent variable by matching it with the first differential equation given in (7). \(u d v+\left(v+u v-u e^{u}\right) \ d u=0\) ; in v ; in u Text Transcription: udv + (v + uv - ue^u) du = 0

In Problems 11–14 verify that the indicated function is an explicit solution of the given differential equation. Assume an appropriate interval I of definition for each solution \(2 y^{\prime}+y=0\) ; \(y=e^{-x / 2}\) Text Transcription: 2y^prime + y = 0 y = e^-x/2

In Problems 11–14 verify that the indicated function is an explicit solution of the given differential equation. Assume an appropriate interval I of definition for each solution \(y^{\prime \prime}+y=\tan x\) ; \(y=-(\cos x) \ln (\sec x+\tan x)\) Text Transcription: y^prime prime + y = tan x y = - (cos x) ln (sec x + tan x)

In Problems 11–14 verify that the indicated function is an explicit solution of the given differential equation. Assume an appropriate interval I of definition for each solution \(y^{\prime \prime}-6 y^{\prime}+13 y=0\) ; \(y=e^{3 x} \cos 2 x\) Text Transcription: y^prime prime - 6y^prime + 13y = 0 y = e^3x cos 2x

In Problems 19 and 20 verify that the indicated expression is an implicit solution of the given first-order differential equation. Find at least one explicit solution \(y=\phi(x)\) in each case. Use a graphing utility to obtain the graph of an explicit solution. Give an interval I of definition of each solution \(\phi\) . \(\frac{d X}{d t}=(X-1)(1-2 X)\) ; \(\ln \left(\frac{2 X-1}{X-1}\right)=t\) Text Transcription: y=phi(x) phi dX/dt = (X-1)(1-2X) ln (2X-1/X-1) = t

In Problems 15–18 verify that the indicated function \(y=\phi(x)\) is an explicit solution of the given first-order differential equation. Proceed as in Example 2, by considering \(\phi\) simply as a function, give its domain. Then by considering \(\phi\) as a solution of the differential equation, give at least one interval I of definition \(y^{\prime}=25+y^{2}\) ; y=5 tan 5x Text Transcription: y=phi(x) phi y^prime = 25 + y^2

In Problems 15–18 verify that the indicated function \(y=\phi(x)\) is an explicit solution of the given first-order differential equation. Proceed as in Example 2, by considering \(\phi\) simply as a function, give its domain. Then by considering \(\phi\) as a solution of the differential equation, give at least one interval I of definition \((y-x) y^{\prime}=y-x+8\) ; \(y=x+4 \sqrt{x+2}\) Text Transcription: y=phi(x) phi (y-x) y^prime = y - x + 8 y = x + 4 sqrt x+2

In Problems 21–24 verify that the indicated family of functions is a solution of the given differential equation. Assume an appropriate interval I of definition for each solution \(\frac{d P}{d t}=P(1-P)\) ; \(P=\frac{c_{1} e^{t}}{1+c_{1} e^{t}}\) Text Transcription: dP/dt = P(1-P) P = c_1 e^t/1 + c_1 e^t

In Problems 15–18 verify that the indicated function \(y=\phi(x)\) is an explicit solution of the given first-order differential equation. Proceed as in Example 2, by considering \(\phi\) simply as a function, give its domain. Then by considering \(\phi\) as a solution of the differential equation, give at least one interval I of definition \(2 y^{\prime}=y^{3} \cos x\) ; \(y=(1-\sin x)^{-1 / 2}\) Text Transcription: y=phi(x) phi 2y^prime = y^3 cos x y=(1-sin x)^-1/2

In Problems 21–24 verify that the indicated family of functions is a solution of the given differential equation. Assume an appropriate interval I of definition for each solution \(\frac{d y}{d x}+2 x y=1\) ; \(y=e^{-x^{2}} \int_{0}^{x} e^{t^{2}} \ d t+c_{1} e^{-x^{2}}\) Text Transcription: dy/dx + 2xy = 1 y = e^-x^2 int_0^x e^t^2 dt + c_1 e^-x^2

In Problems 19 and 20 verify that the indicated expression is an implicit solution of the given first-order differential equation. Find at least one explicit solution \(y=\phi(x)\) in each case. Use a graphing utility to obtain the graph of an explicit solution. Give an interval I of definition of each solution \(\phi\) . \(2 x y \ d x+\left(x^{2}-y\right) \ d y=0\) ; \(-2 x^{2} y+y^{2}=1\) Text Transcription: y=phi(x) phi 2xy dx + (x^2 - y) dy = 0 -2x^2 y + y^2 = 1

In Problems 21–24 verify that the indicated family of functions is a solution of the given differential equation. Assume an appropriate interval I of definition for each solution \(\frac{d^{2} y}{d x^{2}}-4 \frac{d y}{d x}+4 y=0\) ; \(y=c_{1} e^{2 x}+c_{2} x e^{2 x}\) Text Transcription: d^2 y/dx^2 - 4dy/dx + 4y = 0 y = c_1 e^2x + c_2 xe^2x

In Problems 15–18 verify that the indicated function \(y=\phi(x)\) is an explicit solution of the given first-order differential equation. Proceed as in Example 2, by considering \(\phi\) simply as a function, give its domain. Then by considering \(\phi\) as a solution of the differential equation, give at least one interval I of definition \(y^{\prime}=2 x y^{2}\) ; \(y=1 /\left(4-x^{2}\right)\) Text Transcription: y=phi(x) phi y^prime = 2xy^2 y=1/(4-x^2)

In Problems 21–24 verify that the indicated family of functions is a solution of the given differential equation. Assume an appropriate interval I of definition for each solution \(x^{3} \frac{d^{3} y}{d x^{3}}+2 x^{2} \frac{d^{2} y}{d x^{2}}-x \frac{d y}{d x}+y=12 x^{2}\) ; \(y=c_{1} x^{-1}+c_{2} x+c_{3} x \ln x+4 x^{2}\) Text Transcription: x^3 d^3 y/dx^3 + 2x^2 d^2 y/dx^2 - x dy/dx + y = 12x^2 y = c_1 x^-1 + c_2 x + c_3 x ln x + 4x^2

Verify that the piecewise-defined function \(y=\left\{\begin{aligned}-x^{2}, & \ \ x<0 \\x^{2}, & \ \ x \geq 0\end{aligned}\right.\) is a solution of the differential equation \(x y^{\prime}-2 y=0\) on \((-\infty, \infty)\). Text Transcription: y = {-x^2, x<0 over x^2, x geq 0 xy^prime - 2y = 0 (-infty, infty)

In Problems 27–30 find values of m so that the function \(y=e^{m x}\) is a solution of the given differential equation. \(y^{\prime}+2 y=0\) Text Transcription: y = e^mx y^prime + 2y = 0

In Example 5 we saw that \(y=\phi_{1}(x)=\sqrt{25-x^{2}}\) and \(y={ }_{2}(x)=-\sqrt{25-x^{2}}\) are solutions of dy/dx=-x/y on the interval (-5, 5). Explain why the piecewise defined function \(y=\left\{\begin{array}{lr}\sqrt{25-x^{2}}, & -5<x<0 \\-\sqrt{25-x^{2}}, & 0 \leq x<5\end{array}\right.\) is not a solution of the differential equation on the interval (-5, 5). Text Transcription: y=phi_1(x)=25-x^2 y=_2(x)=-sqrt 25-x^2 y={ sqrt 25-x^2, -5<x<0 over -sqrt 25-x^2, 0 leq x<5

In Problems 27–30 find values of m so that the function \(y=e^{m x}\) is a solution of the given differential equation. \(5 y^{\prime}=2 y\) Text Transcription: y = e^mx 5y^prime=2y

In Problems 27–30 find values of m so that the function \(y=e^{m x}\) is a solution of the given differential equation. \(y^{\prime \prime}-5 y^{\prime}+6 y=0\) Text Transcription: y = e^mx y^prime prime - 5y^prime + 6y = 0

In Problems 31 and 32 find values of m so that the function \(y=x^{m}\) is a solution of the given differential equation. \(x^{2} y^{\prime \prime}-7 x y^{\prime}+15 y=0\) Text Transcription: y=x^m x^2 y^prime prime - 7xy^prime + 15y = 0

In Problems 31 and 32 find values of m so that the function \(y=x^{m}\) is a solution of the given differential equation. \(x y^{\prime \prime}+2 y^{\prime}=0\) Text Transcription: y=x^m xy^prime prime + 2y^prime = 0

In Problems 33–36 use the concept that y = c, \(-\infty<x<\infty\), is a constant function if and only if \(y^{\prime}=0\) to determine whether the given differential equation possesses constant solutions. \(3 x y^{\prime}+5 y=10\) Text Transcription: (-infty<x<infty) y^prime=0 3xy^prime+5y=10

In Problems 33–36 use the concept that y = c, \(-\infty<x<\infty\), is a constant function if and only if \(y^{\prime}=0\) to determine whether the given differential equation possesses constant solutions. \(y^{\prime}=y^{2}+2 y-3\) Text Transcription: (-infty<x<infty) y^prime=0 y^prime = y^2 + 2y - 3

In Problems 33–36 use the concept that y = c, \(-\infty<x<\infty\), is a constant function if and only if \(y^{\prime}=0\) to determine whether the given differential equation possesses constant solutions. \((y-1) y^{\prime}=1\) Text Transcription: (-infty<x<infty) y^prime=0 (y-1) y^prime=1

In Problems 33–36 use the concept that y = c, \(-\infty<x<\infty\), is a constant function if and only if \(y^{\prime}=0\) to determine whether the given differential equation possesses constant solutions. \(y^{\prime \prime}+4 y^{\prime}+6 y=10\) Text Transcription: (-infty<x<infty) y^prime=0 y^prime prime + 4y^prime + 6y = 10

In Problems 27–30 find values of m so that the function \(y=e^{m x}\) is a solution of the given differential equation. \(2 y^{\prime \prime}+7 y^{\prime}-4 y=0\) Text Transcription: y = e^mx 2y^prime prime + 7y^prime - 4y = 0

In Problems 37 and 38 verify that the indicated pair of functions is a solution of the given system of differential equations on the interval \((-\infty, \infty)\). \(\frac{d x}{d t}=x+3 y\) \(\frac{d y}{d t}=5 x+3 y\) ; \(x=e^{-2 t}+3 e^{6 t}\) , \(y=-e^{-2 t}+5 e^{6 t}\) Text Transcription: (-infty, infty) dx/dt=x+3y dy/dt=5x+3y x=e^-2t+3e^6t y=-e^-2t+5e^{6t

In Problems 37 and 38 verify that the indicated pair of functions is a solution of the given system of differential equations on the interval \((-\infty, \infty)\). \(\frac{d^{2} x}{d t^{2}}=4 y+e^{t}\) \(\frac{d^{2} y}{d t^{2}}=4 x-e^{t}\) ; \(x=\cos 2 t+\sin 2 t+\frac{1}{5} e^{t}\) , \(y=-\cos 2 t-\sin 2 t-\frac{1}{5} e^{t}\) Text Transcription: (-infty, infty) d^2 x/dt^2 = 4y + e^t d^2 y/dt^2 = 4x - e^t x=cos 2t + sin 2t + 1/5 e^t y = -cos 2t - sin 2t - 1/5 e^t

Make up a differential equation that you feel confident possesses only the trivial solution y = 0. Explain your reasoning.

Make up a differential equation that does not possess any real solutions.

What function do you know from calculus is such that its first derivative is itself? Its first derivative is a constant multiple k of itself? Write each answer in the form of a first-order differential equation with a solution.

What function (or functions) do you know from calculus is such that its second derivative is itself? Its second derivative is the negative of itself? Write each answer in the form of a second-order differential equation with a solution.

Discuss why it makes intuitive sense to presume that the linear differential equation \(y^{\prime \prime}+2 y^{\prime}+4 y=5 \sin t\) has a solution of the form y = A sin t + B cos t, where A and B are constants. Then find specific constants A and B so that y = A sin t + B cos t is a particular solution of the DE. Text Transcription: y^prime prime + 2y^prime + 4y = 5 sin t

In Problems 45 and 46 the given figure represents the graph of an implicit solution G(x, y) = 0 of a differential equation dy/dx = f(x, y). In each case the relation G(x, y) = 0 implicitly defines several solutions of the DE. Carefully reproduce each figure on a piece of paper. Use different colored pencils to mark off segments, or pieces, on each graph that correspond to graphs of solutions. Keep in mind that a solution \(\phi\) must be a function and differentiable. Use the solution curve to estimate an interval I of definition of each solution \(\phi\). Text Transcription: phi

The graphs of members of the one-parameter family \(x^{3}+y^{3}=3 c x y\) are called folia of Descartes. Verify that this family is an implicit solution of the first-order differential equation \(\frac{d y}{d x}=\frac{y\left(y^{3}-2 x^{3}\right)}{x\left(2 y^{3}-x^{3}\right)}\). Text Transcription: x^3 + y^3 = 3cxy dy/dx = y(y^3 - 2x^3)/x(2y^3 - x^3)

In Problems 45 and 46 the given figure represents the graph of an implicit solution G(x, y) = 0 of a differential equation dy/dx = f(x, y). In each case the relation G(x, y) = 0 implicitly defines several solutions of the DE. Carefully reproduce each figure on a piece of paper. Use different colored pencils to mark off segments, or pieces, on each graph that correspond to graphs of solutions. Keep in mind that a solution \(\phi\) must be a function and differentiable. Use the solution curve to estimate an interval I of definition of each solution \(\phi\). Text Transcription: phi

The graph in Figure 1.1.7 is the member of the family of folia in Problem 47 corresponding to c = 1. Discuss: How can the DE in Problem 47 help in finding points on the graph of \(x^{3}+y^{3}=3 x y\) where the tangent line is vertical? How does knowing where a tangent line is vertical help in determining an interval I of definition of a solution \(\phi\) of the DE? Carry out your ideas, and compare with your estimates of the intervals in Problem 46. Text Transcription: x^3 + y^3 = 3xy) phi

Discuss, and illustrate with examples, how to solve differential equations of the forms dy/dx = f(x) and \(d^{2} y / d x^{2}=f(x)\). Text Transcription: d^2 y/dx^2 = f(x)

In Problem 21 a one-parameter family of solutions of the DE \(P^{\prime}=P(1-P)\) is given. Does any solution curve pass through the point (0, 3)? Through the point (0, 1)? Text Transcription: P^prime = P(1-P)

The differential equation \(x\left(y^{\prime}\right)^{2}-4 y^{\prime}-12 x^{3}=0\) has the form given in (4). Determine whether the equation can be put into the normal form dy/dx = f(x, y). Text Transcription: x(y^prime)^2 - 4y^prime - 12x^3 = 0

In Example 5 the largest interval I over which the explicit solutions \(y=\phi_{1}(x)\) and \(y=\phi_{2}(x)\) are defined is the open interval (-5, 5). Why can’t the interval I of definition be the closed interval -5, 5]? Text Transcription: y=phi_1(x) y=phi_2(x)

The normal form (5) of an nth-order differential equation is equivalent to (4) whenever both forms have exactly the same solutions. Make up a first-order differential equation for which \(F\left(x, y, y^{\prime}\right)=0\) is not equivalent to the normal form dy/dx = f(x, y). Text Transcription: F(x, y, y^prime) = 0

Given that y = sin x is an explicit solution of the first order differential equation \(\frac{d y}{d x}=\sqrt{1-y^{2}}\) . Find an interval I of definition [Hint: I is not the interval \((-\infty, \infty)\).] Text Transcription: dy/dx=sqrt 1-y^2 (-infty, infty)

Qualitative information about a solution \(y=\phi(x)\) of a differential equation can often be obtained from the equation itself. Before working Problems 55–58, recall the geometric significance of the derivatives dy/dx and \(d^{2} y / d x^{2}\). Consider the differential equation dy/dx = 5 - y. (a) Either by inspection or by the method suggested in Problems 33–36, find a constant solution of the DE. (b) Using only the differential equation, find intervals on the y-axis on which a nonconstant solution \(y=\phi(x)\) is increasing. Find intervals on the y-axis on which \(y=\phi(x)\) is decreasing. Text Transcription: y=phi(x) d^2 y / dx^2

Find a linear second-order differential equation \(F\left(x, y, y^{\prime}, y^{\prime \prime}\right)=0\) for which \(y=c_{1} x+c_{2} x^{2}\) is a two parameter family of solutions. Make sure that your equation is free of the arbitrary parameters \(c_{1}\) and \(c_{2}\). Text Transcription: F(x, y, y^prime, y^prime prime) = 0 y = c_1 x + c_2 x^2 c_1 c_2

Qualitative information about a solution \(y=\phi(x)\) of a differential equation can often be obtained from the equation itself. Before working Problems 55–58, recall the geometric significance of the derivatives dy/dx and \(d^{2} y / d x^{2}\). Consider the differential equation \(d y / d x=e^{-x^{2}}\). (a) Explain why a solution of the DE must be an increasing function on any interval of the x-axis. (b) What are \(\lim _{x \rightarrow-\infty} d y / d x\) and \(\lim _{x \rightarrow \infty} d y / d x\) What does this suggest about a solution curve as \(x \rightarrow \pm \infty\)? (c) Determine an interval over which a solution curve is concave down and an interval over which the curve is concave up. (d) Sketch the graph of a solution \(y=\phi(x)\) of the differential equation whose shape is suggested by parts (a)–(c). Text Transcription: y=phi(x) d^2 y / dx^2 dy/dx = e^-x^2 lim_x rightarrow -infty dy/dx lim_x rightarrow infty dy/dx x rightarrow pm infty

In Problems 59 and 60 use a CAS to compute all derivatives and to carry out the simplifications needed to verify that the indicated function is a particular solution of the given differential equation. \(y^{(4)}-20 y^{\prime \prime \prime}+158 y^{\prime \prime}-580 y^{\prime}+841 y=0\) ; \(y=x e^{5 x} \cos 2 x\) Text Transcription: y^(4) - 20y^prime prime prime + 158y^prime prime - 580y^prime + 841y = 0 y=xe^5x cos 2x

Qualitative information about a solution \(y=\phi(x)\) of a differential equation can often be obtained from the equation itself. Before working Problems 55–58, recall the geometric significance of the derivatives dy/dx and \(d^{2} y / d x^{2}\). Consider the differential equation \(y^{\prime}=y^{2}+4\). (a) Explain why there exist no constant solutions of the DE. (b) Describe the graph of a solution \(y=\phi(x)\). For example, can a solution curve have any relative extrema? (c) Explain why y = 0 is the y-coordinate of a point of inflection of a solution curve (d) Sketch the graph of a solution \(y=\phi(x)\) of the differential equation whose shape is suggested by parts (a)–(c). Text Transcription: y=phi(x) d^2 y / dx^2 y^prime = y^2 + 4

Qualitative information about a solution \(y=\phi(x)\) of a differential equation can often be obtained from the equation itself. Before working Problems 55–58, recall the geometric significance of the derivatives dy/dx and \(d^{2} y / d x^{2}\). Consider the differential equation dy/dx = y(a - by), where a and b are positive constants. (a) Either by inspection or by the method suggested in Problems 33–36, find two constant solutions of the DE. (b) Using only the differential equation, find intervals on the y-axis on which a nonconstant solution \(y=\phi(x)\) is increasing. Find intervals on which \(y=\phi(x)\) is decreasing. (c) Using only the differential equation, explain why y = a/2b is the y-coordinate of a point of inflection of the graph of a non constant solution \(y=\phi(x)\). (d) On the same coordinate axes, sketch the graphs of the two constant solutions found in part (a). These constant solutions partition the xy-plane into three regions. In each region, sketch the graph of a non constant solution \(y=\phi(x)\) whose shape is suggested by the results in parts (b) and (c). Text Transcription: y=phi(x) d^2 y / dx^2